Finding the phase transition for Friedman's long finite sequences
Willem M. Baartse
Abstract:
The phase transition that we will focus on is about fast growing
sequences, which were discovered by Friedman. PA can prove that the
length of these sequences remains finite, but IΣ_2 cannot. IΣ_2 is the
subsystem of PA where the induction scheme is limited to induction of
Σ_2 formulas. This system can prove the totality of a function if and
only if it is multiple recursive. The multiple recursive functions are
the functions that can be defined using the elementary functions and
nested recursion schemes with some finite number of variables. So it
can prove the totality of the primitive recursive functions. It can
also prove the totality of the Ackermann function which is double
recursive. The multiple recursive functions are the same as the <
ω^{ω^ω} recursive functions. Hence, IΣ_2 proves the totality of the
Hardy functions H_α for α < ω^{ω^ω} but does not prove the totality of
H_{ω^{ω^ω}}. In section 4, which follows section 5 from [4], we will
first show that the growth of the length of these sequences is ω^{ω^ω}
recursive (this is a direct consequence of a theorem from [14]), which
implies that IΣ_3 (and thus PA) can prove that the length of these
sequences remains finite. Then, using lemma 3.2.5 it will be shown
that this growth is so fast that it eventually dominates every <
ω^{ω^ω} recursive function. From theorem 3.3.3 it then follows that
IΣ_2 cannot prove that these sequences remain of finite length.
Section 5 is based on an article with Weiermann. In section 5.1 we
show that if the parameter function grows very slowly, it is easy to
find an upper bound on the length of the sequences. In section 5.2 we
make a con- struction which shows that for a slowly growing function f
the length of the sequences doesn’t grow significantly slower than for
the identity function. To nicely characterize the phase transition we
use that IΣ_2 proves the totality of H_α iff α < ω^{ω^ω}. This known
fact is proved in section 3.3 using a theorem from [1] and the notion
of ordinal recursion.
In the last section we study a similar phase transition which seems
easier to characterize. It is again about sequences but now an extra
condition is introduced which makes the sequences grow so fast that PA
is no longer able to prove that they remain finite. The phase
transition here is very similar to the one in section 5.